# Copyright (c) 2017 Weitian LI # MIT license """ Solve the Fokker-Planck equation to derive the time evolution of the electron spectrum (or number density distribution). """ import logging import numpy as np logger = logging.getLogger(__name__) def TDMAsolver(a, b, c, d): """ Tri-diagonal matrix algorithm (a.k.a Thomas algorithm) solver, which is much faster than the generic Gaussian elimination algorithm. a[i]*x[i-1] + b[i]*x[i] + c[i]*x[i+1] = d[i], where: a[0] = c[N-1] = 0 Example ------- >>> A = np.array([[10, 2, 0, 0], [ 3, 10, 4, 0], [ 0, 1, 7, 5], [ 0, 0, 3, 4]], dtype=float) >>> a = np.array([ 3, 1, 3], dtype=float) >>> b = np.array([10, 10, 7, 4], dtype=float) >>> c = np.array([ 2, 4, 5 ], dtype=float) >>> d = np.array([ 3, 4, 5, 6], dtype=float) >>> print(TDMAsolver(a, b, c, d)) [ 0.14877589 0.75612053 -1.00188324 2.25141243] # compare against numpy linear algebra library >>> print(np.linalg.solve(A, d)) [ 0.14877589 0.75612053 -1.00188324 2.25141243] References ---------- [1] http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm Credit ------ [1] https://gist.github.com/cbellei/8ab3ab8551b8dfc8b081c518ccd9ada9 """ # Number of equations nf = len(d) # Copy the input arrays ac, bc, cc, dc = map(np.array, (a, b, c, d)) for it in range(1, nf): mc = ac[it-1] / bc[it-1] bc[it] -= mc*cc[it-1] dc[it] -= mc*dc[it-1] xc = bc xc[-1] = dc[-1] / bc[-1] for il in range(nf-2, -1, -1): xc[il] = (dc[il] - cc[il]*xc[il+1]) / bc[il] return xc class FokkerPlanckSolver: """ Solve the Fokker-Planck equation. ∂u(x,t) ∂ / ∂u(x) \ u(x,t) ------- = -- | B(x)u(x) + C(x)----- | + Q(x,t) - ------ ∂t ∂x \ ∂x / T(x,t) u(x,t) : distribution/spectrum w.r.t. x at different times B(x,t) : advection coefficient C(x,t) : diffusion coefficient (>0) Q(x,t) : injection coefficient (>=0) T(x,t) : escape coefficient References ---------- [1] Park & Petrosian 1996, ApJS, 103, 255 http://adsabs.harvard.edu/abs/1996ApJS..103..255P [2] Donnert & Brunetti 2014, MNRAS, 443, 3564 http://adsabs.harvard.edu/abs/2014MNRAS.443.3564D """ def __init__(self, xmin, xmax, grid_num, buffer_np, tstep, f_advection, f_diffusion, f_injection, f_escape=None): self.xmin = xmin self.xmax = xmax # Number of points on the logarithmic grid self.grid_num = grid_num # Number of grid points for the buffer region near the lower boundary self.buffer_np = buffer_np # Time step self.tstep = tstep # Function f(x,t) to calculate the advection coefficient B(x,t) self.f_advection = f_advection # Function f(x,t) to calculate the diffusion coefficient C(x,t) self.f_diffusion = f_diffusion # Function f(x,t) to calculate the injection coefficient Q(x,t) self.f_injection = f_injection # Function f(x,t) to calculate the escape coefficient T(x,t) self.f_escape = f_escape @property def x(self): """ X values of the adopted logarithmic grid. """ grid = np.logspace(np.log10(self.xmin), np.log10(self.xmax), num=self.grid_num) return grid @property def dx(self): """ Values of dx[i] on the grid. dx[i] = (x[i+1] - x[i-1]) / 2 NOTE: Extrapolate the x grid by 1 point beyond each side, therefore avoid NaN for the first and last element of dx[i]. Otherwise, the following calculation of tridiagonal coefficients may be invalid on the boundary elements. References: Ref.[1],Eq.(8) """ x = self.x # Extrapolate the x grid by 1 point beyond each side x2 = np.concatenate([ [x[0]**2/x[1]], x, [x[-1]**2/x[-2]], ]) dx_ = (x2[2:] - x2[:-2]) / 2 return dx_ @property def dx_phalf(self): """ Values of dx[i+1/2] on the grid. dx[i+1/2] = x[i+1] - x[i] Thus the last element is NaN. References: Ref.[1],Eq.(8) """ x = self.x dx_ = x[1:] - x[:-1] grid = np.concatenate([dx_, [np.nan]]) return grid @property def dx_mhalf(self): """ Values of dx[i-1/2] on the grid. dx[i-1/2] = x[i] - x[i-1] Thus the first element is NaN. """ x = self.x dx_ = x[1:] - x[:-1] grid = np.concatenate([[np.nan], dx_]) return grid @staticmethod def X_phalf(X): """ Calculate the values at midpoints (+1/2) for the given quantity. X[i+1/2] = (X[i] + X[i+1]) / 2 Thus the last element is NaN. References: Ref.[1],Eq.(10) """ Xmid = (X[1:] + X[:-1]) / 2 return np.concatenate([Xmid, [np.nan]]) @staticmethod def X_mhalf(X): """ Calculate the values at midpoints (-1/2) for the given quantity. X[i-1/2] = (X[i-1] + X[i]) / 2 Thus the first element is NaN. """ Xmid = (X[1:] + X[:-1]) / 2 return np.concatenate([[np.nan], Xmid]) @staticmethod def W(w): # References: Ref.[1],Eqs.(27,35) with np.errstate(invalid="ignore"): # Ignore NaN's w = np.abs(w) mask = (w < 0.1) # Comparison on NaN gives False, as expected W = np.zeros(w.shape) * np.nan W[mask] = 1.0 / (1 + w[mask]**2/24 + w[mask]**4/1920) W[~mask] = (w[~mask] * np.exp(-w[~mask]/2) / (1 - np.exp(-w[~mask]))) return W @staticmethod def bound_w(w, wmin=1e-8, wmax=1e3): """ Bound the absolute values of w within [wmin, wmax]. To avoid the underflow/overflow during later W/Wplus/Wminus calculations. """ with np.errstate(invalid="ignore"): # Ignore NaN's m1 = (np.abs(w) < wmin) m2 = (np.abs(w) > wmax) ww = np.array(w) ww[m1] = wmin * np.sign(ww[m1]) ww[m2] = wmax * np.sign(ww[m2]) return ww def Wplus(self, w): # References: Ref.[1],Eq.(32) ww = self.bound_w(w) W = self.W(ww) Wplus = W * np.exp(ww/2) return Wplus def Wminus(self, w): # References: Ref.[1],Eq.(32) ww = self.bound_w(w) W = self.W(ww) Wminus = W * np.exp(-ww/2) return Wminus def tridiagonal_coefs(self, tc, uc): """ Calculate the coefficients for the tridiagonal system of linear equations corresponding to the original Fokker-Planck equation. -a[i]*u[i-1] + b[i]*u[i] - c[i]*u[i+1] = r[i], where: a[0] = c[N-1] = 0 NOTE ---- When i=0 or i=N-1, b[i] is invalid due to X[-1/2] or X[N-1/2] are invalid. Therefore, b[0] and b[N-1] should be alternatively calculated with (e.g., no-flux) boundary condition considered. References: Ref.[1],Eqs.(16,18,34) """ x = self.x dx = self.dx dx_phalf = self.dx_phalf dx_mhalf = self.dx_mhalf dt = self.tstep B = np.array([self.f_advection(x_, tc) for x_ in x]) C = np.array([self.f_diffusion(x_, tc) for x_ in x]) Q = np.array([self.f_injection(x_, tc) for x_ in x]) # B_phalf = self.X_phalf(B) B_mhalf = self.X_mhalf(B) C_phalf = self.X_phalf(C) C_mhalf = self.X_mhalf(C) w_phalf = dx_phalf * B_phalf / C_phalf w_mhalf = dx_mhalf * B_mhalf / C_mhalf Wplus_phalf = self.Wplus(w_phalf) Wplus_mhalf = self.Wplus(w_mhalf) Wminus_phalf = self.Wminus(w_phalf) Wminus_mhalf = self.Wminus(w_mhalf) # a = (dt/dx) * (C_mhalf/dx_mhalf) * Wminus_mhalf a[0] = 0.0 # Fix a[0] which is NaN c = (dt/dx) * (C_phalf/dx_phalf) * Wplus_phalf c[-1] = 0.0 # Fix c[-1] which is NaN b = 1 + (dt/dx) * ((C_mhalf/dx_mhalf) * Wplus_mhalf + (C_phalf/dx_phalf) * Wminus_phalf) # Calculate b[0] & b[-1], considering the no-flux boundary condition b[0] = 1 + (dt/dx[0]) * (C_phalf[0]/dx_phalf[0])*Wminus_phalf[0] b[-1] = 1 + (dt/dx[-1]) * (C_mhalf[-1]/dx_mhalf[-1])*Wplus_mhalf[-1] # Escape from the system if self.f_escape is not None: T = np.array([self.f_escape(x_, tc) for x_ in x]) b += dt / T # Right-hand side r = dt * Q + uc return (a, b, c, r) def fix_boundary(self, uc): """ Truncate the lower end (i.e., near the lower boundary) of the distribution/spectrum and then extrapolate as a power law, in order to avoid the unphysical pile-up of electrons at the lower regime. References: Ref.[2],Sec.(3.3) """ uc = np.asarray(uc) x = self.x # Calculate the power-law index xa = x[self.buffer_np] xb = x[self.buffer_np+1] ya = uc[self.buffer_np] yb = uc[self.buffer_np+1] if ya > 0 and yb > 0: # Truncate and extrapolate as a power law s = np.log(yb/ya) / np.log(xb/xa) uc[:self.buffer_np] = ya * (x[:self.buffer_np] / xa) ** s return uc def solve_step(self, tc, uc): """ Solve the Fokker-Planck equation by a single step. """ a, b, c, r = self.tridiagonal_coefs(tc=tc, uc=uc) TDM_a = -a[1:] # Also drop the first element TDM_b = b TDM_c = -c[:-1] # Also drop the last element TDM_rhs = r t2 = tc + self.tstep u2 = TDMAsolver(TDM_a, TDM_b, TDM_c, TDM_rhs) u2 = self.fix_boundary(u2) # Clear negative number densities # u2[u2 < 0] = 0 return (t2, u2) def solve(self, u0, tstart, tstop): """ Solve the Fokker-Planck equation from ``tstart`` to ``tstop``, with initial spectrum/distribution ``u0``. """ uc = u0 tc = tstart logger.info("Solving Fokker-Planck equation: " + "time: %.3f - %.3f" % (tstart, tstop)) nstep = (tstop - tc) / self.tstep i = 0 while tc < tstop: i += 1 logger.debug("[%d/%d] t=%.3f ..." % (i, nstep, tc)) tc, uc = self.solve_step(tc, uc) return uc